Subgroups — #31012656

Order 17152.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22572558	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22562433	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562391	C₂	1	2	yes	maximal, complemented
5	#22562346	C₂	1	1	yes	maximal, characteristic, complemented
6	#22562294	C₂	1	2	yes	maximal, complemented
7	#22562244	C₂	1	1	yes	maximal, characteristic, complemented
8	#22562119	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#357×D₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₆₇	C₂²	1	2	yes	—
13	#16864051	—	2	1	no	—
14	D₄×D₂₆₈	—	2	1	no	—
15	C₂×#14523984	C₂²	1	2	yes	complemented
16	C₂×C₁₆×D₆₇	C₂²	1	2	yes	complemented
17	#16863801	C₂²	1	1	yes	characteristic, complemented
18	#16863773	C₂²	1	1	yes	characteristic
19	#16863708	—	2	2	no	—
20	#16863676	—	2	1	no	—
21	#16863633	C₂²	1	2	yes	—
22	#16863629	C₂²	1	2	yes	complemented
23	#16863591	—	2	1	no	—
24	#16863531	—	2	2	no	—
25	C₈×D₂₆₈	C₂²	1	1	yes	characteristic
26	#107×D₆₇	C₂²	1	2	yes	complemented
27	#16863221	C₂²	1	2	yes	complemented
28	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
29	#16863198	C₂²	1	2	yes	complemented
30	#16863198	C₂²	1	2	yes	complemented
31	#16863197	C₂²	1	2	yes	—
32	#16863196	C₂²	1	2	yes	complemented
33	(C₂×D₆₇)⋊C₁₆	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#366×C₆₇	C₂²	1	1	yes	characteristic, complemented
38	#357×C₆₇	C₂²	1	1	yes	characteristic
39	#357×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
41	C₂×#11761628	—	4	1	no	—
42	C₂²×D₂₆₈	—	4	1	no	—
43	C₂×D₄×D₆₇	—	2	2	no	—
44	C₂×D₅₃₆	—	2	1	no	—
45	C₂×D₅₃₆	C₂³	1	2	yes	—
46	C₂×#11761613	—	2	2	no	—
47	D₈×D₆₇	—	2	4	no	—
48	D₈×D₆₇	—	4	2	no	—
49	C₂×C₈×D₆₇	C₂³	1	2	yes	—
50	#14524043	—	2	1	no	—
51	#14524040	—	2	1	no	—
52	#14524025	—	4	2	no	—
53	#14524022	—	4	1	no	—
54	#14524018	—	2	2	no	—
55	#14524017	—	4	1	no	—
56	#14524014	—	2	2	no	—
57	#14524013	—	4	2	no	—
58	C₄×D₂₆₈	D₄	1	1	yes	characteristic
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	#14523984	—	4	2	no	—
61	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
62	C₁₆×D₆₇	—	4	2	no	—
63	#14523977	—	2	1	no	—
64	#14523977	C₂³	1	2	yes	—
65	#14523976	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	D₄	1	1	yes	characteristic
67	#14523964	—	2	2	no	—
68	#14523963	—	2	2	no	—
69	(C₂×D₆₇)⋊C₈	D₄	1	2	yes	—
70	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₆₇	—	2	1	no	—
73	C₄×D₄×C₆₇	—	2	1	no	—
74	D₁₆×C₆₇	—	2	2	no	—
75	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
77	#107×C₆₇	C₂³	1	1	yes	characteristic
78	#107×C₆₇	C₂³	1	1	yes	characteristic
79	#102×C₆₇	—	2	1	no	—
80	#102×C₆₇	—	2	1	no	—
81	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×D₆₇	—	4	2	no	—
83	#11761628	—	8	4	no	—
84	C₂×Dic₁₃₄	—	4	1	no	—
85	C₂×D₂₆₈	—	2	2	no	—
86	C₂×D₂₆₈	—	4	2	no	—
87	C₂×D₂₆₈	—	4	1	no	—
88	C₂×D₂₆₈	—	4	1	no	—
89	C₂×D₂₆₈	—	8	4	no	—
90	C₂×D₂₆₈	D₈	1	1	yes	characteristic
91	C₂×C₂²⋊D₆₇	—	4	2	no	—
92	C₂×C₂²⋊D₆₇	—	4	2	no	—
93	D₄×D₆₇	—	4	4	no	—
94	D₄×D₆₇	—	8	2	no	—
95	C₂×C₄×D₆₇	—	4	2	no	—
96	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
97	D₅₃₆	—	2	2	no	—
98	D₅₃₆	—	2	2	no	—
99	D₅₃₆	—	8	1	no	—
100	#11761613	—	4	4	no	—
101	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
102	C₈×D₆₇	—	2	2	no	—
103	C₈×D₆₇	—	2	2	no	—
104	C₄⋊Dic₆₇	—	2	2	no	—
105	C₄⋊Dic₆₇	—	4	1	no	—
106	C₄⋊Dic₆₇	D₈	1	1	yes	characteristic
107	Dic₆₇⋊C₄	—	4	2	no	—
108	C₂²⋊Dic₆₇	—	4	2	no	—
109	(C₂×D₆₇)⋊C₄	—	4	2	no	—
110	(C₂×D₆₇)⋊C₄	D₈	1	2	yes	—
111	C₆₇⋊C₁₆	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	4	1	no	—
115	C₂²×C₄×C₆₇	—	4	1	no	—
116	D₈×C₆₇	—	2	1	no	—
117	D₈×C₆₇	—	2	1	no	—
118	D₈×C₆₇	—	2	1	no	—
119	D₈×C₆₇	#41	1	2	yes	complemented
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	—	2	1	no	—
123	C₄⋊C₄×C₆₇	—	2	1	no	—
124	C₂²⋊C₄×C₆₇	—	4	1	no	—
125	C₂²⋊C₄×C₆₇	—	4	1	no	—
126	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
127	C₁₆×C₆₇	—	2	1	no	—
128	C₁₆×C₆₇	—	2	1	no	—
129	C₂³×C₆₇	—	4	1	no	—
130	C₂³×C₆₇	—	4	1	no	—
131	C₂²×D₆₇	—	4	2	no	—
132	C₂²×D₆₇	—	8	4	no	—
133	C₂²×D₆₇	—	8	4	no	—
134	C₂²×D₆₇	—	16	2	no	—
135	C₂²×D₆₇	C₂×D₈	1	2	yes	—
136	D₄×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	4	1	no	—
138	D₄×C₆₇	—	4	1	no	—
139	D₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	—	8	2	no	—
143	C₂×C₄×C₆₇	—	8	2	no	—
144	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₆₇	—	8	4	no	—
148	C₂²⋊D₆₇	—	8	4	no	—
149	C₂×Dic₆₇	—	4	2	no	—
150	C₂×Dic₆₇	C₂×D₈	1	2	yes	—
151	D₂₆₈	—	4	2	no	—
152	D₂₆₈	—	4	2	no	—
153	D₂₆₈	—	8	4	no	—
154	D₂₆₈	—	8	2	no	—
155	D₂₆₈	—	16	1	no	—
156	D₂₆₈	D₁₆	1	4	yes	complemented
157	C₄×D₆₇	—	2	2	no	—
158	C₄×D₆₇	—	2	2	no	—
159	C₄×D₆₇	—	8	4	no	—
160	Dic₁₃₄	—	8	2	no	—
161	C₈×C₆₇	—	2	1	no	—
162	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
163	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
164	C₆₇⋊C₈	—	2	2	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	8	1	no	—
167	C₂²×C₆₇	—	8	1	no	—
168	C₂²×C₆₇	—	8	1	no	—
169	C₂²×C₆₇	—	8	1	no	—
170	C₂²×C₆₇	—	8	1	no	—
171	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
172	C₂×D₆₇	—	2	2	no	—
173	C₂×D₆₇	—	2	2	no	—
174	C₂×D₆₇	—	8	2	no	—
175	C₂×D₆₇	—	8	2	no	—
176	C₂×D₆₇	—	8	2	no	—
177	C₂×D₆₇	—	16	8	no	—
178	C₂×D₆₇	C₂×D₁₆	1	4	yes	—
179	C₄×C₆₇	—	2	1	no	—
180	C₄×C₆₇	—	8	1	no	—
181	C₄×C₆₇	—	8	1	no	—
182	C₄×C₆₇	C₂×D₁₆	1	2	yes	—
183	C₄×C₆₇	#546	1	1	yes	characteristic
184	C₄×C₆₇	#577	1	1	yes	characteristic
185	Dic₆₇	—	2	2	no	—
186	Dic₆₇	—	8	2	no	—
187	#33861	—	67	1	no	maximal, Sylow
188	C₂×C₆₇	—	8	2	no	—
189	C₂×C₆₇	—	16	1	no	—
190	C₂×C₆₇	#3294	1	1	yes	characteristic
191	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
192	C₂×C₆₇	#3416	1	1	yes	characteristic
193	D₆₇	—	2	4	no	—
194	D₆₇	—	16	2	no	—
195	#3334	—	67	1	no	—
196	D₄×D₈	—	67	1	no	—
197	#2240	—	67	1	no	—
198	#2232	—	67	2	no	—
199	#2224	—	67	1	no	—
200	#2215	—	67	2	no	—
201	#2206	—	67	1	no	—
202	#2184	—	67	2	no	—
203	C₄×D₁₆	D₆₇	1	1	yes	characteristic, complemented
204	D₄×C₁₆	—	67	1	no	—
205	C₂×#357	—	67	2	no	—
206	C₆₇	#33861	1	1	yes	characteristic, complemented, Sylow
207	C₂²×D₈	—	67	2	no	—
208	#547	—	134	1	no	—
209	D₄²	—	134	1	no	—
210	C₂×D₁₆	—	67	2	no	—
211	C₂×D₁₆	C₂×D₆₇	1	1	yes	characteristic
212	C₂²×C₁₆	—	67	2	no	—
213	#500	—	67	1	no	—
214	#493	—	67	1	no	—
215	#480	—	134	2	no	—
216	#474	—	134	1	no	—
217	#466	—	67	2	no	—
218	#466	—	67	2	no	—
219	#459	—	134	1	no	—
220	#447	—	134	2	no	—
221	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic
222	C₄×D₈	C₂×D₆₇	1	1	yes	characteristic, complemented
223	C₈×D₄	—	67	1	no	—
224	C₂×#107	—	67	2	no	—
225	#366	—	67	2	no	—
226	#366	C₂×D₆₇	1	1	yes	characteristic, complemented
227	C₄⋊C₁₆	—	67	1	no	—
228	#357	—	67	2	no	—
229	#357	—	67	2	no	—
230	#357	—	67	2	no	—
231	#357	—	67	2	no	—
232	#357	C₂×D₆₇	1	1	yes	characteristic
233	#357	C₂×D₆₇	1	1	yes	characteristic, complemented
234	C₂²⋊C₁₆	—	67	2	no	—
235	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
236	C₂×#41	—	268	1	no	—
237	C₂²×D₄	—	268	1	no	—
238	C₂²×D₄	—	134	2	no	—
239	C₂×D₈	—	268	2	no	—
240	C₂×D₈	—	67	2	no	—
241	C₂×D₈	—	134	2	no	—
242	C₂×D₈	—	134	4	no	—
243	C₂×D₈	—	134	1	no	—
244	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×D₈	C₂²×D₆₇	1	1	yes	characteristic
246	C₂²×C₈	—	67	2	no	—
247	C₄⋊Q₈	—	134	1	no	—
248	#127	—	134	1	no	—
249	#123	—	268	2	no	—
250	C₂²⋊Q₈	—	268	1	no	—
251	#121	—	268	1	no	—
252	#121	—	134	2	no	—
253	#121	—	134	2	no	—
254	#120	—	268	2	no	—
255	C₄×D₄	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	67	1	no	—
258	C₂×C₄⋊C₄	—	134	2	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	268	2	no	—
261	C₂×C₁₆	—	268	2	no	—
262	C₂×C₁₆	—	67	2	no	—
263	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
264	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
265	#107	—	67	2	no	—
266	#107	—	134	2	no	—
267	#107	—	134	1	no	—
268	#107	C₂²×D₆₇	1	1	yes	characteristic
269	#107	C₂²×D₆₇	1	1	yes	characteristic
270	C₄⋊C₈	—	67	1	no	—
271	#102	—	2	1	no	—
272	#102	—	2	1	no	—
273	#102	—	134	2	no	—
274	#102	—	134	2	no	—
275	C₂²⋊C₈	—	67	2	no	—
276	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
277	C₂⁴	—	268	2	no	—
278	#41	—	536	4	no	—
279	C₂×Q₈	—	268	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	268	2	no	—
283	C₂×D₄	—	268	2	no	—
284	C₂×D₄	—	268	2	no	—
285	C₂×D₄	—	268	4	no	—
286	C₂×D₄	—	268	1	no	—
287	C₂×D₄	—	268	1	no	—
288	C₂×D₄	—	536	4	no	—
289	C₂×D₄	—	536	2	no	—
290	C₂×D₄	—	67	1	no	—
291	C₂×D₄	—	134	2	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	4	1	no	—
294	C₂²×C₄	—	268	2	no	—
295	C₂²×C₄	—	67	2	no	—
296	D₈	—	2	1	no	—
297	D₈	—	2	1	no	—
298	D₈	—	2	1	no	—
299	D₈	—	268	4	no	—
300	D₈	—	536	1	no	—
301	D₈	—	134	2	no	—
302	D₈	—	134	2	no	—
303	D₈	#11761628	1	2	yes	complemented
304	C₂×C₈	—	67	2	no	—
305	C₂×C₈	—	134	2	no	—
306	C₂×C₈	—	134	2	no	—
307	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	268	2	no	—
312	C₄⋊C₄	—	268	1	no	—
313	C₄⋊C₄	—	67	1	no	—
314	C₄⋊C₄	—	134	2	no	—
315	C₂²⋊C₄	—	4	1	no	—
316	C₂²⋊C₄	—	4	1	no	—
317	C₂²⋊C₄	—	268	2	no	—
318	C₂²⋊C₄	—	268	2	no	—
319	C₂²⋊C₄	—	67	2	no	—
320	C₄²	D₄×D₆₇	1	1	yes	characteristic
321	C₁₆	—	2	1	no	—
322	C₁₆	—	2	1	no	—
323	C₁₆	—	134	2	no	—
324	C₂³	—	4	1	no	—
325	C₂³	—	4	1	no	—
326	C₂³	—	268	2	no	—
327	C₂³	—	536	4	no	—
328	C₂³	—	536	4	no	—
329	C₂³	—	1072	2	no	—
330	C₂³	—	67	2	no	—
331	Q₈	—	536	2	no	—
332	D₄	—	2	2	no	—
333	D₄	—	4	1	no	—
334	D₄	—	4	1	no	—
335	D₄	—	4	1	no	—
336	D₄	—	268	2	no	—
337	D₄	—	268	2	no	—
338	D₄	—	536	4	no	—
339	D₄	—	536	4	no	—
340	D₄	—	536	4	no	—
341	D₄	—	536	2	no	—
342	D₄	—	1072	1	no	—
343	D₄	—	67	4	no	—
344	C₂×C₄	—	4	1	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	8	2	no	—
347	C₂×C₄	—	8	2	no	—
348	C₂×C₄	—	268	2	no	—
349	C₂×C₄	—	536	4	no	—
350	C₂×C₄	—	67	2	no	—
351	C₂×C₄	—	134	2	no	—
352	C₂×C₄	—	134	2	no	—
353	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
354	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	134	2	no	—
358	C₈	C₂×#11761628	1	1	yes	characteristic
359	C₈	#14524089	1	1	yes	characteristic
360	C₂²	—	4	2	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	8	1	no	—
363	C₂²	—	8	1	no	—
364	C₂²	—	8	1	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	536	2	no	—
367	C₂²	—	536	2	no	—
368	C₂²	—	536	2	no	—
369	C₂²	—	1072	8	no	—
370	C₂²	—	67	4	no	—
371	C₂²	—	134	2	no	—
372	C₂²	—	134	2	no	—
373	C₂²	C₂×D₈×D₆₇	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	536	2	no	—
378	C₄	—	134	2	no	—
379	C₄	D₁₆×D₆₇	1	2	yes	—
380	C₄	#16864042	1	1	yes	characteristic
381	C₄	#16864203	1	1	yes	characteristic
382	C₂	—	8	2	no	—
383	C₂	—	16	1	no	—
384	C₂	—	1072	2	no	—
385	C₂	—	134	4	no	—
386	C₂	#22572024	1	1	yes	characteristic
387	C₂	C₂×D₁₆×D₆₇	1	1	yes	characteristic
388	C₂	#22573352	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₆₇
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₄×D₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₈
hypercenter	C₄×D₁₆	D₆₇
hypocenter	C₆₇	#33861
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22572558	C₂	semidirect
2. D₈×D₂₆₈	C₂	semidirect
3. #22562433	C₂	semidirect
4. #22562391	C₂	semidirect
5. #22562346	C₂	semidirect
6. #22562294	C₂	semidirect
7. #22562244	C₂	semidirect
8. #22562119	C₂	semidirect
9. C₁₆×D₂₆₈	C₂	semidirect
10. #357×D₆₇	C₂	semidirect
11. C₄×D₁₆×C₆₇	C₂	semidirect
15. C₂×#14523984	C₂²	semidirect
16. C₂×C₁₆×D₆₇	C₂²	semidirect
17. #16863801	C₂²	semidirect
22. #16863629	C₂²	semidirect
26. #107×D₆₇	C₂²	semidirect
27. #16863221	C₂²	semidirect
28. C₄⋊(C₆₇⋊C₁₆)	C₂²	semidirect
29. #16863198	C₂²	semidirect
30. #16863198	C₂²	semidirect
32. #16863196	C₂²	semidirect
33. (C₂×D₆₇)⋊C₁₆	C₂²	semidirect
36. C₄×D₈×C₆₇	C₂²	semidirect
37. #366×C₆₇	C₂²	semidirect
39. #357×C₆₇	C₂²	semidirect
40. C₄×C₁₆×C₆₇	C₂²	semidirect
119. D₈×C₆₇	#41	semidirect
156. D₂₆₈	D₁₆	semidirect
203. C₄×D₁₆	D₆₇	semidirect
206. C₆₇	#33861	semidirect
222. C₄×D₈	C₂×D₆₇	semidirect
226. #366	C₂×D₆₇	semidirect
233. #357	C₂×D₆₇	semidirect
235. C₄×C₁₆	C₂×D₆₇	semidirect
303. D₈	#11761628	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22572558	2
D₈×D₂₆₈	2
#22562433	2
#22562391	2
#22562346	2
#22562294	2
#22562244	2
#22562119	2
C₁₆×D₂₆₈	2
#357×D₆₇	2
C₄×D₁₆×C₆₇	2
#33861	67